Stepwise Method for Drawing Parallel Lines Using Simple Tools
We are basically going to learn how to construct parallel lines and the steps to construct parallel lines.
What is A Parallel Line?
Parallel lines are lines which do not have a common meeting point in the same plane, however far they are extended .The following figure represents parallel lines. Sometimes you may be presented with one line and you need to create another line parallel to it through any given point. You might think to simply take a straight edge and draw a line that might seem right; however, you could not be sure that the line you constructed is actually parallel. With the use of geometry and a compass, you can easily plot additional points that will ensure the line you construct is truly parallel.
Properties of Parallel Lines
1) The corresponding angles formed by parallel lines are equal.
2) The vertically opposite angles formed by parallel lines are equal.
3) The alternate interior angles formed by parallel lines are equal.
4) The alternate exterior angles formed by parallel lines are equal.
5) The pair of interior angles on the same side of the transversal are supplementary, that is they equal to 180 degrees.
Construction of Parallel Lines:
Given: We have been a point P .
Construct : We have to construct parallel lines.
The Steps for Constructing Parallel Lines Is Quite Simple! Here Are The Steps To Construct Parallel Lines-
Step 1) Use your straightedge, and draw a transversal through the given point . This is simply a straight line which passes through the given point P and intersects with the given line. Drawing the line slanted will make the construction easier than to try to make the line in a vertical manner. Be sure that you draw the line properly above point P.
Step 2) Using the construction you can copy an angle, construct a copy of the angle formed by the transversal and the given line such that the copy will be located UP at point P. The vertex of the copied angle will be located at the point P.
Step 3) When you draw the line to complete the angle copy, you will be able to draw a line parallel to the given line.
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Steps To Construct Parallel Lines
How To Construct Parallel Lines?
Let’s understand Step by Step to Construct Parallel Lines-
If we have to construct a line parallel to the other line from an external point all we require is a ruler and a compass and the following steps need to be kept in mind!
Given: A line segment named AB and a given point P that lies out of the line segment AB.
To construct a line that is parallel to line AB that passes through the given point P.
Step 1: You have to choose any point X on the given line segment AB and join it to point P as shown below in the diagram.
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Step 2: Considering X as the center and any suitable radius you need to draw an arc cutting the line segment PX at the point M and AB at point N respectively.
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Step 3: As you have P as the center and radius remains same as used in the previous step 2 you need to draw an arc EF cutting the line segment PX at point Q as shown in the figure below.
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Step 4: With Q as the center and same radius as we have used in Step 1, draw an arc cutting the arc EF at R as shown below in the given diagram.
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Step 5: Join the points R and P to draw a line segment CD as shown in the figure given below.
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The line segment CD is the line that needs to be parallel to the line segment AB and passes through the point P.
Note: Now you might think how to know whether a line is parallel to the other given line or not. To check whether or not two lines are parallel, we must compare their slopes which are denoted by m in the equation (y = mx+c). Two lines are parallel if and only if their slopes which are m are equal in measure.
FAQs on How to Construct Parallel Lines from an External Point
1. What are the main steps to construct a line parallel to a given line through an external point using a ruler and compass?
To construct a parallel line, you can use the property of equal alternate interior angles. The steps are as follows:
- Step 1: Draw a line (say, AB) and mark a point (P) outside it.
- Step 2: Take any point (Q) on line AB and join it to P, creating a transversal line PQ.
- Step 3: With Q as the centre and any convenient radius, draw an arc that cuts AB at X and PQ at Y.
- Step 4: With P as the centre and the same radius, draw another arc that cuts PQ at Z.
- Step 5: Use the compass to measure the distance between X and Y.
- Step 6: With Z as the centre and the measured distance (XY) as the radius, cut the arc drawn from P. Name this point R.
- Step 7: Join P and R and extend the line. This new line is parallel to AB because you have constructed a pair of equal alternate interior angles.
2. What are the essential geometric tools required for the construction of parallel lines?
For constructing parallel lines accurately as per the CBSE/NCERT syllabus, you primarily need two essential tools from your geometry box:
- A Ruler (or a straightedge): This is used for drawing the initial line, the transversal, and the final parallel line.
- A Compass: This is crucial for drawing arcs and accurately copying angles or measuring distances to ensure the new line is perfectly parallel to the original.
A sharpened pencil is also necessary for making precise markings.
3. Can you explain the concept behind constructing parallel lines? Why does this method work?
The method of constructing parallel lines using a ruler and compass works because it relies on the geometric properties of angles formed when a transversal intersects two lines. The most common method is based on the converse of the alternate interior angles theorem. This theorem states that if a transversal cuts two lines in such a way that the pair of alternate interior angles is equal, then the two lines are parallel. By using a compass to measure and replicate an angle at the external point, you are essentially forcing the alternate interior angles to be equal, which guarantees that the line you draw is parallel.
4. Is it possible to construct parallel lines using set squares instead of a compass?
Yes, it is possible and often quicker to draw parallel lines using a pair of set squares or one set square and a ruler. The method is as follows:
- Place a ruler or one set square on the paper and hold it firmly.
- Place the other set square alongside the ruler's edge.
- Draw a line along one of the perpendicular edges of the set square.
- Now, slide the set square along the ruler's edge to a new position.
- Draw another line along the same edge of the set square.
The two lines you have drawn will be parallel to each other.
5. What is the importance of drawing parallel lines in geometry and in real-world applications?
The ability to construct parallel lines is a fundamental skill in geometry and has significant real-world importance. In geometry, it is the basis for constructing various shapes like parallelograms, trapezoids, and rectangles. In the real world, the concept is applied in:
- Architecture and Construction: To ensure walls are parallel, floors are level, and in designing structures like buildings and bridges.
- Art and Design: Artists use parallel lines to create perspective and depth in drawings.
- Engineering: For designing railway tracks, roads, and circuit boards where components must be aligned perfectly.
6. What is the key difference between constructing a parallel line and a perpendicular line from an external point?
The key difference lies in the angle relationship you create.
- When constructing a parallel line, your goal is to copy an existing angle (like an alternate interior or corresponding angle) to ensure the lines have the same orientation and never intersect.
- When constructing a perpendicular line, your goal is to create a new line that intersects the original line at a perfect 90-degree angle. The construction steps for this involve creating arcs that bisect a line segment to find the point directly opposite the external point.
In short, one aims for zero intersection (parallel), while the other aims for a 90° intersection (perpendicular).
7. What is the primary property of parallel lines that distinguishes them from intersecting lines?
The primary property of parallel lines is that they lie in the same plane and maintain a constant distance from each other at all points. Because of this constant separation, they will never intersect, no matter how far they are extended in either direction. In contrast, intersecting lines cross each other at a single, unique point. This non-intersecting nature is the defining characteristic of parallel lines.
